Problem 18 - Applied Physics

Question

If the kinetic energy of an object is doubled, by what factor is its velocity increased?

Step-by-Step Solution

Answer

The velocity is increased by a factor of \( \sqrt{2} \), or approximately 1.414.

Step 1: Understand Kinetic Energy Formula

The formula for kinetic energy (KE) is given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. We want to understand how the velocity changes when the kinetic energy is doubled.

Step 2: Set Up Equation for Doubled Kinetic Energy

If the kinetic energy is doubled, the new kinetic energy can be written as \( 2KE = 2 \times \frac{1}{2}mv^2 = mv^2 \). Let \( v' \) be the new velocity. For the new kinetic energy, you could write \( mv'^2 = 2(\frac{1}{2}mv^2) = mv^2 \).

Step 3: Solve for the New Velocity

Since \( mv'^2 = mv^2 \), you can divide both sides by \( m \) (assuming \( m eq 0 \)) to get \( v'^2 = 2v^2 \). Take the square root of both sides to find \( v' = \sqrt{2}v \).

Step 4: Determine the Increase Factor

The velocity \( v' \) is \( \sqrt{2} \) times the original velocity \( v \). Hence, the factor by which the velocity increases is \( \sqrt{2} \), which is approximately 1.414.

Key Concepts

Velocity

Velocity is a fundamental concept in physics, describing the speed of an object in a specific direction. It is a vector, meaning it has both magnitude and direction. In contexts like kinetic energy, velocity is key because it directly influences how much energy an object has while moving.

To understand velocity fully, think about how speed and direction work together. Speed is how fast something is going, while velocity tells us both the speed and its direction, like 50 km/h north.

• If a car moves in a straight line at a constant speed, its velocity is constant.
• A change in speed or a change in direction results in a change in velocity.

Velocity is crucial in calculations involving kinetic energy because it is squared in the kinetic energy formula.

Kinetic Energy Formula

The kinetic energy formula is a mathematical way to calculate the energy that an object possesses due to its motion. The formula is given by:\[ KE = \frac{1}{2}mv^2 \]Where:

• \( KE \) is the kinetic energy,
• \( m \) is the mass of the object, and
• \( v \) is the velocity of the object.

This equation highlights how both mass and velocity impact an object's kinetic energy. Importantly, velocity is squared, showing its significant influence. If velocity doubles, kinetic energy increases by a factor of four, showing the relationship's quadratic nature.

Understanding the formula helps in solving problems that ask us to manipulate kinetic energy, such as calculating changes in velocity when kinetic energy changes.

Doubled Kinetic Energy

When we say kinetic energy is doubled, it means the existing energy level of an object is multiplied by two. For example, if an object's kinetic energy was initially 100 Joules, doubling it increases to 200 Joules.

This doubling impacts its velocity due to the kinetic energy formula. If kinetic energy changes, velocity must change to adjust for this increase, assuming mass remains constant. This relationship gives rise to calculations to find new velocities based on changes in kinetic energy.

• Keep in mind that while mass is constant, the velocity's change is not straightforward due to its squared value in the formula.

Doubling kinetic energy challenges us to adjust the other variables in the kinetic energy equation, mainly affecting velocity calculations.

Velocity Increase Factor

The velocity increase factor answers the question: by what proportion does velocity change when kinetic energy is modified? When kinetic energy is doubled, the task is to find how this affects velocity.

Based on the formula, when kinetic energy doubles:\[ v'^2 = 2v^2 \]This equation illustrated that velocity \( v' \) changes by a factor of \( \sqrt{2} \). Solving this gives:\[ v' = \sqrt{2}v \]Thus, the velocity is scaled up by \( \sqrt{2} \), approximately 1.414, meaning the velocity increases by about 41.4%.

This velocity increase factor is key in many physics applications, demonstrating how kinetic energy and velocity interplay. It highlights the non-linear relationship as velocity change doesn't equate to a simple doubling when kinetic energy doubles.